The Trajectory of a Projectile

12 Key excerpts on "The Trajectory of a Projectile"

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  eBook - PDF

  University Physics Volume 1

  • William Moebs, Samuel J. Ling, Jeff Sanny(Authors)
  • 2016(Publication Date)
  • Openstax

    (Publisher)

  • Find the time of flight and impact velocity of a projectile that lands at a different height from that of launch. • Calculate The Trajectory of a Projectile. Projectile motion is the motion of an object thrown or projected into the air, subject only to acceleration as a result of gravity. The applications of projectile motion in physics and engineering are numerous. Some examples include meteors as they enter Earth’s atmosphere, fireworks, and the motion of any ball in sports. Such objects are called projectiles and their path is called a trajectory. The motion of falling objects as discussed in Motion Along a Straight Line is a simple one-dimensional type of projectile motion in which there is no horizontal movement. In this section, we consider two- dimensional projectile motion, and our treatment neglects the effects of air resistance. The most important fact to remember here is that motions along perpendicular axes are independent and thus can be analyzed separately. We discussed this fact in Displacement and Velocity Vectors, where we saw that vertical and horizontal motions are independent. The key to analyzing two-dimensional projectile motion is to break it into two motions: one along the horizontal axis and the other along the vertical. (This choice of axes is the most sensible because acceleration resulting from gravity is vertical; thus, there is no acceleration along the horizontal axis when air resistance is negligible.) As is customary, we call the horizontal axis the x-axis and the vertical axis the y-axis. It is not required that we use this choice of axes; it is simply convenient in the case of gravitational acceleration. In other cases we may choose a different set of axes. Figure 4.11 illustrates the notation for displacement, where we define s → to be the total displacement, and x → and y → are its component vectors along the horizontal and vertical axes, respectively.

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  Sports Biomechanics

  The Basics: Optimising Human Performance

  • Prof. Anthony J. Blazevich(Author)
  • 2017(Publication Date)
  • Bloomsbury Sport

    (Publisher)

  CHAPTER 3 PROJECTILE MOTION

  What is the optimum angle of trajectory or flight path (that is, the angle thrown relative to the ground) for a shot-putter aiming to throw the maximum distance? (Hint: not 45°.) What factors affect maximum throwing distance and to what degree?

  By the end of this chapter you should be able to:

  • List the factors that influence an object’s trajectory

  • Use the equations of projectile motion to calculate flight times, ranges and projection angles of projectiles

  • Design a simple model to determine the influence of factors affecting projection range

  • Create a spreadsheet to speed up calculations to optimise athletic throwing performance

  • Complete a video analysis of a throw to optimise performance

  Projectile motion refers to the motion of an object (for example a shot, ball or human body) projected at an angle into the air. Gravity and air resistance affect such objects, although in many cases air resistance is considered to be so small that it can be disregarded. A projected object can move at any angle between horizontal (0°) and vertical (90°) but gravity only acts on bodies moving with some vertical motion.

  Trajectory is influenced by the projection speed, the projection angle and the relative height of projection (that is, the vertical distance between the landing and release points; for example, in a baseball throw that lands on the ground, the vertical distance is the height above the ground from which the ball was released).

  FIG. 3.1 Tennis ball trajectory. Gravity accelerates the ball towards the ground at the same rate regardless of whether the tennis player leaves the ball to fall freely or hits it perfectly horizontally. However, the trajectory of the ball is different in these two circumstances.

  Projection speed

  The distance a projectile covers, its range, is chiefly influenced by its projection speed. The faster the projection speed, the further the object will go. If an object is thrown through the air, the distance it travels before hitting the ground (its range) will be a function of horizontal velocity and flight time (that is, velocity × time, as you saw in Chapter 1 ). In Figure 3.1

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  Physics

  • John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler(Authors)
  • 2015(Publication Date)
  • Wiley

    (Publisher)

  3.3 Projectile Motion Projectile motion is an idealized kind of motion that occurs when a moving object (the projectile) experiences only the acceleration due to gravity, which acts vertically down- ward. If the trajectory of the projectile is near the earth’s surface, the vertical component a y of the acceleration has a magnitude of 9.80 m/s 2 . The acceleration has no horizontal component (a x 5 0 m/s 2 ), the effects of air resistance being negligible. There are several symmetries in projectile motion: (1) The time to reach maximum height from any vertical level is equal to the time spent returning from the maximum height to that level. (2) The speed of a projectile depends only on its height above its launch point, and not on whether it is moving upward or downward. 3.4 Relative Velocity The velocity of object A relative to object B is v B AB , and the velocity of object B relative to object C is v B BC . The velocity of A relative to C is shown in Equation 3 (note the ordering of the subscripts). While the velocity of object A relative to object B is v B AB , the velocity of B relative to A is v B AB 5 2 v B AB . v B 5 r B 2 r B 0 t 2 t 0 5 D r B Dt (3.1) v B 5 lim Dt B0 D r B D t (1) a B 5 v B 2 v B 0 t 2 t 0 5 Dv B D t (3.2) B a 5 lim Dt B0 D v B Dt (2) v B AC 5 v B AB 1 v B BC (3) x Component v x 5 v 0x 1 a x t (3.3a) x 5 1 2 (v 0x 1 v x )t (3.4a) x 5 v 0x t 1 1 2 a x t 2 (3.5a) v x 2 5 v 0x 2 1 2a x x (3.6a) y Component v y 5 v 0y 1 a y t (3.3b) y 5 1 2 (v 0y 1 v y )t (3.4b) y 5 v 0y t 1 1 2 a y t 2 (3.5b) v y 2 5 v 0y 2 1 2a y y (3.6b) FOCUS ON CONCEPTS Note to Instructors: The numbering of the questions shown here reflects the fact that they are only a representative subset of the total number that are available online. However, all of the questions are available for assignment via an online homework management program such as WileyPLUS or WebAssign.

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  Physics

  • John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler(Authors)
  • 2021(Publication Date)
  • Wiley

    (Publisher)

  3.3 Projectile Motion Projectile motion is an idealized kind of motion that occurs when a moving object (the projectile) experiences only the acceleration due to gravity, which acts vertically downward. If the trajectory of the projectile is near the earth’s surface, the ver- tical component a y of the acceleration has a magnitude of 9.80 m/s 2 . The acceleration has no horizontal component (a x = 0 m/s 2 ), the effects of air resistance being negligible. There are several symmetries in projectile motion: (1) The time to reach maximum height from any vertical level is equal to the time spent returning from the maximum height to that level. (2) The speed of a projectile depends only on its height above its launch point, and not on whether it is moving upward or downward. 3.4 Relative Velocity The velocity of object A relative to object B is → v AB , and the velocity of object B relative to object C is → v BC . The velocity of A relative to C is shown in Equation 3 (note the ordering of the subscripts). While the velocity of object A relative to object B is → v AB , the velocity of B relative to A is → v BA = − → v AB . → v AC = → v AB + → v BC (3) Focus on Concepts Additional questions are available for assignment in WileyPLUS. Section 3.3 Projectile Motion 1. The drawing shows projectile motion at three points along the trajectory. The speeds at the points are υ 1 , υ 2 , and υ 3 . Assume there is no air resistance and rank the speeds, largest to smallest. (Note that the symbol > means “greater than.”) (a) υ 1 > υ 3 > υ 2 (b) υ 1 > υ 2 > υ 3 (c) υ 2 > υ 3 > υ 1 (d) υ 2 > υ 1 > υ 3 (e) υ 3 > υ 2 > υ 1 2. Two balls are thrown from the top of a building, as in the draw- ing. Ball 1 is thrown straight down, and ball 2 is thrown with the same speed, but upward at an angle θ with respect to the horizontal. Consider the motion of the balls after they are released.

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  Physics

  • John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler(Authors)
  • 2018(Publication Date)
  • Wiley

    (Publisher)

  3.3 Projectile Motion Projectile motion is an idealized kind of motion that occurs when a moving object (the projectile) experiences only the ac- celeration due to gravity, which acts vertically downward. If the trajectory of the projectile is near the earth’s surface, the vertical component a y of the acceleration has a magnitude of 9.80 m/s 2 . The acceleration has no horizontal component (a x = 0 m/s 2 ), the effects of air resistance being negligible. There are several symmetries in projectile motion: (1) The time to reach maximum height from any vertical level is equal to the time spent returning from the maximum height to that level. (2) The speed of a projectile depends only on its height above its launch point, and not on whether it is moving upward or downward. 3.4 Relative Velocity The velocity of object A relative to object B is v → AB , and the velocity of object B relative to object C is v → BC . The velocity of A rel- ative to C is shown in Equation 3 (note the ordering of the subscripts). While the velocity of object A relative to object B is v → AB , the velocity of B relative to A is v → BA = −v → AB . v → AC = v → AB + v → BC (3) Note to Instructors: The numbering of the questions shown here reflects the fact that they are only a representative subset of the total number that are available online. However, all of the questions are available for assignment via WileyPLUS. Section 3.3 Projectile Motion 1. The drawing shows projectile motion at three points along the trajectory. The speeds at the points are υ 1 , υ 2 , and υ 3 . Assume there is no air resistance and rank the speeds, largest to smallest. (Note that the symbol > means “greater than.”) (a) υ 1 > υ 3 > υ 2 (b) υ 1 > υ 2 > υ 3 (c) υ 2 > υ 3 > υ 1 (d) υ 2 > υ 1 > υ 3 (e) υ 3 > υ 2 > υ 1 3. Two balls are thrown from the top of a building, as in the drawing. Ball 1 is thrown straight down, and ball 2 is thrown with the same speed, but upward at an angle θ with respect to the horizontal.

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  The Flight of Uncontrolled Rockets

  International Series of Monographs on Aeronautics and Astronautics

  • F. R. Gantmakher, L. M. Levin, H. L. Dryden(Authors)
  • 2014(Publication Date)
  • Pergamon

    (Publisher)

  CHAPTER 2 CALCULATION OF THE TRAJECTORY OF A ROCKET 7. Statement of the Problem The external ballistics of conventional projectiles (shell) may be divided into what are known as the basic problem and the special problems. The basic problem in the external ballistics of shell is concerned with the motion of a particle (coinciding with the shell's mass-centre) under the action of two forces, namely, the force of gravity exerted vertically downwards, and the resistance of the air in the direction opposite to that of the shell's velocity. Here (1) the earth and atmosphere are regarded as fixed, (2) the surface of the earth is assumed flat and coincident with the plane of the horizon at the launch point, and (3) the acceleration g of the force of gravity is assumed constant. The trajectory of a shell is calculated and its basic elements defined on these assumptions. Thus, for example, given the initial velocity v 0 , the quadrant elevation (Q.E.)0 o , and the ballistic coefficient c, we may determine the total horizontal range X, the velocity v c and the trajectory inclination 0 C at the point of impact, and the height of the trajectory Y. In the basic problem we ignore the rotational motion of the shell, the aero-dynamic lift force perpendicular to the velocity, the effect of wind, etc. The effects of these additional factors are studied as special problems involving cor-rections to the trajectory elements determined in the basic problem. The trajectory of an uncontrolled rocket comprises two sections, namely, that of powered flight and that of unpowered or coasting flight. Calculations for the unpowered section can be made by the usual methods employed in the basic problem of external ballistics. This requires knowledge of the initial data: the velocity vector v a at the commencement of unpowered flight (i.e. at the end of powered flightf), having magnitude v & and making an angle 0 a with the horizontal, and the ballistic coefficient c.

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  A Certain Uncertainty

  Nature's Random Ways

  • Mark P. Silverman(Author)
  • 2014(Publication Date)
  • Cambridge University Press

    (Publisher)

  7 On target: uncertainties of projectile flight A projectile, while it moves by a motion compounded of a uniform horizontal motion and a motion naturally accelerated downward, describes under this movement a semi-parabolic line. —Galileo Galilei 1 7.1 Knowing where they come down In his satirical song 2 about WW II German rocket engineer Werner von Braun, songwriter and erstwhile mathematician Tom Lehrer sings (in a German accent): Vonce ze rocketts are up, who cares vere zey come down? Physicists, of course, care very much where they come down. Indeed, the study of projectile motion is ordinar- ily a fundamental part of any study of classical dynamics, introductory or advanced, where it serves primarily to illustrate the laws of motion applied to objects in free-fall in a uniform gravitational field. In this context, as evidenced by numerous textbooks – beginning with Galileo’s own, first published in 1638 – students are taught to solve problems that fall into certain standard categories such as (1) ground-to-ground targeting (e.g. a missile is fired with speed v at an angle θ to the horizontal at a target a horizontal distance d away), (2) air-to-ground targeting (e.g. a package is dropped at a height h above the ground from an airplane traveling horizontally at speed v), (3) ground-to-air targeting (e.g. a projectile is launched at speed v and angle θ to the horizontal at a pie plate simultaneously dropped from a height h and horizontal distance d), and possibly others. In the commonly encountered textbook and classroom examples the specified dynamical variables are exact, rather than distributed, quantities, and the objective 1 Galileo Galilei, from Discorsi e Dimostrazioni Matematiche Intorno a Due Nuove Scienze [Discourses and Mathematical Demonstrations relating to Two New Sciences] (Elzevir, 1638), Fourth Day: Theorem I, Proposition I. (Translation from Italian by M P Silverman) 2 Tom Lehrer song, “Werner von Braun”, performed in 1965 in San Francisco CA.

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  Guide to Mechanics

  • Philip Dyke, Roger Whitworth(Authors)
  • 2017(Publication Date)
  • Red Globe Press

    (Publisher)

  If the velocity of projection is V at an angle to Ox , as illustrated in Figure 5.12, then the velocity of projection can be written as the vector: u ˆ V cos i ‡ V sin j The horizontal motion is unaffected by the acceleration and the component of velocity in that direction is constant. The upwards vertical motion is subject to an acceleration of g . The acceleration vector is written in the form: a ˆ g j For a given point r relative to O at time t on the trajectory, we can write: r ˆ u t ‡ 1 2 a t 2 The vector diagram in Figure 5.13 shows the dependence of the position vector r on the vectors u and a . This gives, for a given point on the path with coordinates ( x , y ): x i ‡ y j ˆ V cos † t i ‡ V sin † t j † ‡ 1 2 g j † t 2 O y V x Figure 5.12 The trajectory 124 Guide to Mechanics O r u t ½ a t 2 Figure 5.13 A triangle of vectors illustrating the equation r ˆ u t ‡ 1 2 a t 2 The set of equations: x ˆ V cos † t and y ˆ V sin † t 1 2 gt 2 represents the parametric equations of the trajectory. Making t the subject of the first of these formulae, we obtain: t ˆ x V cos This enables us to derive: y ˆ x tan gx 2 sec 2 2 V 2 5 : 26 † as the Cartesian equation for the trajectory. From this equation, we can see that, for a given V and , the trajectory is a parabola. Having once determined this we can use any of the properties of a parabola to discuss projectile motion. The most important of these is the symmetry of the curve about a vertical line through its maximum value, which is of course its greatest height. Using this symmetry property, we can make the following observations about projectile motion: (a) The greatest height is the maximum value of y . (b) The range is the value of x for which y ˆ 0. (c) The range is twice the x value to the greatest height. (The time to the range is twice the time to the greatest height, which follows directly.) (d) All heights are symmetrical about the greatest height's horizontal position.

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  Introduction to Physics

  • John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler(Authors)
  • 2015(Publication Date)
  • Wiley

    (Publisher)

  Check Your Understanding (The answers are given at the end of the book.) 3. A projectile is fired into the air, and it follows the parabolic path shown in the drawing, land- ing on the right. There is no air resistance. At any instant, the projectile has a velocity v B and an acceleration a B . Which one or more of the drawings could not represent the directions for v B and a B at any point on the trajectory? (a) (b) (c) (d) v v v a a a a v The range in the previous example depends on the angle u at which the projectile is fired above the horizontal. When air resistance is absent, the maximum range results when u 5 458. In projectile motion, the magnitude of the acceleration due to gravity affects the trajectory in a significant way. For example, a baseball or a golf ball would travel much farther and higher on the moon than on the earth, when launched with the same initial velocity. The reason is that the moon’s gravity is only about one-sixth as strong as the earth’s. Section 2.6 points out that certain types of symmetry with respect to time and speed are present for freely falling bodies. These symmetries are also found in projectile motion, since projectiles are falling freely in the vertical direction. In particular, the time required for a projectile to reach its maximum height H is equal to the time spent returning to the ground. In addition, Figure 3.11 shows that the speed v of the object at any height above the ground on the upward part of the trajectory is equal to the speed v at the same height on the downward part. Although the two speeds are the same, the velocities are different, because they point in different directions. Conceptual Example 9 shows how to use this type of symmetry in your reasoning. CONCEPTUAL EXAMPLE 9 | Two Ways to Throw a Stone From the top of a cliff overlooking a lake, a person throws two stones.

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  Theoretical Mechanics for Sixth Forms

  In Two Volumes

  • C. Plumpton, W. A. Tomkys(Authors)
  • 2016(Publication Date)
  • Pergamon

    (Publisher)

  *4.4. The Parabolic Path of a Projectile Equation (4.14) shows that the path of a projectile, projected under the conditions we have assumed throughout this chapter, is a parabola. We now obtain the equation of the path referred to (downward) vertical and horizontal axes Ox and Oy through the highest point of the path (Fig. 4.6). The particle is projected from A with velocity u at an angle a with the horizontal and P(x, y) is the point of the path reached by the particle at time t from the instant when the particle was at O. Then x = Ygt 2 , y — ut cos a so that the equation of the path is 2t/ 2 xcos 2 a ,. ,_. y 2 = . (4.17) g The path of the projectile is thus seen to be a parabola with latus rectum 2w 2 cos 2 α/g, focus at (w 2 cos 2 a/2g, 0), and directrix x = — u 2 cos 2 a/2g. The length of the latus rectum is a function of the horizontal component of the velocity of projection. From equation (4.8) the height of O above A is u 2 sin 2 a/2g; therefore Λ , . , _ t . A . u 2 sin 2 a u 2 cos a u 2 the height of the directrix above A is — 1 = — . 2g 2g 2g 94 THEORETICAL MECHANICS FIG. 4.6 Thus the height of the directrix above the point of projection is inde-pendent of the angle of projection of the particle and is equal to the height which the particle would reach if it were thrown vertically upwards from A. It follows that the parabolic paths of all particles projected from A with speed u have the same directrix. When the particle is at a point Q whose distance below the directrix is d, the horizontal component of its velocity is u cos a and the vertical component of its velocity is |Äp sin2a -2 *(IH}· Therefore the speed of the particle at Q is /{-2 cos 2 a + u 2 sin 2 a — 2g GHI -V(2gd). This speed is equal to the speed that would be acquired by the particle in falling from the directrix to Q. Example. Three parallel vertical walls of heights A, H, h are at equal distances a apart.

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  Engineering Mechanics

  Dynamics

  • L. G. Kraige, J. N. Bolton(Authors)
  • 2018(Publication Date)
  • Wiley

    (Publisher)

  Therefore, every- thing covered in Art. 2 ∕ 2 on rectilinear motion can be applied separately to the x-motion and to the y-motion. Projectile Motion An important application of two-dimensional kinematic theory is the problem of projectile motion. For a first treatment of the subject, we neglect aerodynamic drag and the curvature and rotation of the earth, and we assume that the altitude change is small enough so that the acceleration due to gravity can be considered constant. With these assumptions, rectangular coordinates are useful for the tra- jectory analysis. For the axes shown in Fig. 2 ∕ 8, the acceleration components are a x = 0 a y = −g r = x i + y j y v = r ˙ = x ˙ i + y ˙ j y a = v ˙ = r ¨ = x ¨ i + y ¨ j y ¨ Path j i x i y j r A x y A v a v y v x a x a y  FIGURE 2/7 Article 2/4 Rectangular Coordinates (x-y) 29 Integration of these accelerations follows the results obtained previously in Art. 2 ∕ 2a for constant acceleration and yields v x = ( v x ) 0 v y = ( v y ) 0 − gt x = x 0 + ( v x ) 0 t y = y 0 + ( v y ) 0 t − 1 2 gt 2 v y 2 = ( v y ) 0 2 − 2 g( y − y 0 ) In all these expressions, the subscript zero denotes initial conditions, frequently taken as those at launch where, for the case illustrated, x 0 = y 0 = 0. Note that the quantity g is taken to be positive throughout this text. We can see that the x- and y-motions are independent for the simple projectile conditions under consideration. Elimination of the time t between the x- and y- displacement equations shows the path to be parabolic (see Sample Problem 2 ∕ 6). If we were to introduce a drag force which depends on the speed squared (for exam- ple), then the x- and y-motions would be coupled (interdependent), and the trajec- tory would be nonparabolic.

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  Sports Math

  An Introductory Course in the Mathematics of Sports Science and Sports Analytics

  • Roland B. Minton(Author)
  • 2016(Publication Date)
  • CRC Press

    (Publisher)

  Chapter 1 Projectile Motion Introduction Basketball star Stephen Curry launches a 3-point shot. As the ball traces its high arc toward the basket, fans rise to their feet in anticipation. Will it go in? Is it a little short? Similar tension accompanies a Jordan Spieth tee shot, an Andy Murray passing shot, a long football pass by Peyton Manning or Li-onel Messi, or a long fly ball by Mike Trout. We will analyze the flights of balls in this chapter as we explore the area of physics known as mechanics. Along the way, we will answer such ques-tions as: How does Blake Griffin hang in the air when dunking? What is the optimal angle to shoot a free throw? Why do golf balls have dimples? Does a knuckleball really dance? The answers are to be found in the funda-mentals of physics. Figuring with Newton Sir Isaac Newton (1643-1727) constructed a framework for the analysis of objects in motion. The second of his three Laws of Motion is the launching point for most of our investigations in this chapter. The shorthand version of Newton’s Second Law is F = ma where F is the sum of all forces acting on an object, m is the object’s mass, and a is the acceleration of the object. One of the most remarkable aspects of 1 2 Sports Math Newton’s Second Law is that it can also be written as F = m a , where F and a appear in bold to indicate that they are multidimensional vector quantities. We will return to this form of the equation when we look at motion in two and three dimensions. The mass m is a scalar (real number) that is related to weight: for earthbound sports, weight is approximately equal to mass times the gravitational constant g . To keep it simple, let’s start with one-dimensional motion; vertical motion, to be precise. In this case, the object’s position can be tracked by its height h above some reference point (e.g., the ground). We define velocity as the rate of change of position with respect to time.

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